Yes, you can. Many people are not advised of this. Keep in mind that if you are using a bow to hunt during muzzleloader or rifle season you must still abide by the state hunter orange requirement and have any applicable permits. You may want to check and make sure there are no special rules regarding felons if you apply to hunt on any of the state's military bases.

I don't practice in TN, but under federal law you're only prohibited from owning/using a firearm.

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Yes you can. And in fact, check with ATF and Tennessee TBI as it may be possible to hunt using black powder muzzle loaders as well if the rifle you use is one allowed by ATF as legally not considered a firearm. http://www.atf.gov/firearms/faq/collectors.html

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